This invention relates to a method for calculating the molecular diffusivity of a solute in a channel, such as a capillary, tube or microfluidic channel. More particularly, this invention relates to calculating molecular diffusivity of a solute using Taylor dispersion concepts.
Recent efforts have been directed towards the development of microscale assay methods in which various chemical and biological processes may be examined in rapid succession and with small amounts of material. Such microscale assay methods can be carried out in, for example, microfluidic devices. A typical microfluidic device, which can be fabricated from a glass, silica or plastic substrate, contains a network of microscale channels through which fluids and chemicals are moved in order to perform an assay. These devices use minute quantities of fluids or other materials, controllably flowed and/or directed, to generate highly reproducible and rapidly changeable microenvironments for control of chemical and biological reaction conditions, enzymatic processes, etc.
Microfluidic devices use small volumes of material. A plug containing the material of interest, such as a molecule (e.g. a protein or DNA molecule), compound, or biological compound is introduced into a microscale channel and observed at least at some point along the channel. Several plugs of a variety of compounds are typically introduced into the same channel, the various plugs being separated by sufficient solvent or buffer material to distinguish adjacent plugs. However, as a plug of material moves along a channel, the material of interest in the plug tends to disperse from the plug into adjacent volumes of solvent or buffer that separate the plug of material from adjacent sample plugs. Such dispersion results from the laminar or parabolic velocity profile of a plug of material in a channel coupled with the molecular diffusivity of the particular material within a particular solvent or buffer. Due to dispersion, a plug of material having a certain length and a certain concentration upon entering a channel will have a longer length and be less concentrated after it travels through the channel.
One advantage of microfluidic devices is that a large variety of small plugs can be introduced and monitored within a channel in rapid succession. The more frequently plugs of material are directed into a channel, the more tests can be run in a smaller amount of time. If the plugs of material are introduced too closely, however, dispersion may cause the solute in one sample plug to overlap the solute in a second adjacent sample plug by the time the plugs travel to the opposite end of the channel. Typically, a detector mechanism is placed at this opposite end to measure a property of each plug of material. Thus, it is helpful to be able to adequately predict how the length of a plug will increase due to dispersion to maximize throughput (i.e., a maximum number of different samples plugs introduced to the channel in a minimum amount of time) and minimize cross-contamination of adjacent sample plugs. If one can predict the amount of dispersion, one can modify the dimensions of the microfluidic device or the process parameters to maximize throughput. One method for maximizing throughput that depends on an accurate prediction of dispersion is discussed in U.S. Pat. No. 6,150,119, which is incorporated herein by reference. In order to accurately predict the dispersion of a sample material, it is necessary to accurately determine the molecular diffusivity of that sample material.
The mathematics of diffusion and dispersion in long thin channels is well understood. Sir Geoffery Taylor developed a method to determine molecular diffusion based on the mass flux in a capillary tube. Taylor's methods are discussed in Taylor, Sir Geoffery, F. R. S. “Conditions of soluble matter in solvent flowing slowly through a tube,” Proceedings of the Royal Society of London, Series A, 219, 186–203 (1953), and Taylor, Sir Geoffrey, F. R. S., “Conditions under which dispersion of a solute in a stream of solvent can be used to measure molecular diffusion,” Proceedings of the Royal Society of London, Series A, 225, 473–477 (1954), both of which are incorporated herein by reference. In particular, Taylor determined that the mass flux along the length of a capillary tube is a sum of convection forces and molecular diffusion. Although dispersion results from a combination of convection and diffusion, the same type of equations that describe diffusion alone can describe the phenomenon of dispersion. Rutherford Aris developed a formula based on the work of Taylor for calculating a dispersion coefficient K. The derivation of that formula is disclosed in Aris, R., “On the dispersion of a solute in a fluid flowing through a tube,” Proceedings of the Royal Society of London, Series A, 235, 67–77 (1956), which is incorporated herein by reference. The formula for the Taylor-Aris dispersion coefficient in a channel with a circular cross-section is:
  K  =      D    (          1      +                                    U            2                    ⁢                      r            2                                    48          ⁢                      D            2                                )  where U is the mean velocity of the sample plug, r is the radius of the capillary tube, and D is the coefficient of molecular diffusion. This formula must be modified to predict the dispersion coefficient in channels with non-circular cross-sections. The following formula was developed for calculating the dispersion coefficient in a channel with a rectangular cross-section:
  K  =      D    (          1      +                        1          210                ⁢                  f          (                      d            w                    )                ⁢                                            U              2                        ⁢                          d              2                                            D            2                                )  where the function ƒ(d/w) is a known function of the depth d and width w of the channel, U is the average velocity and D is the molecular diffusivity. A detailed derivation of this formula is in Chatwin, P. C. and P. J. Sullivan, “The effect of aspect ratio on longitudinal diffusivity in rectangular channels,” Journal of Fluid Mechanics, 120,347–358 (1982), which is incorporated herein by reference.
When a sample plug consisting of a sharp pulse of material is inserted into a stream of fluid flowing through a channel, the concentration profile of the plug will change as it travels down the length of the channel. Material will disperse both in front of and behind the boundaries of the original plug of material. Accordingly, the concentration profile of the material observed at a point downstream of the injection will not be bounded by sudden increases in concentration. Instead, the observed concentration profile will start off at a small concentration as the beginning of the plug crosses the detection point, increasing to a peak as more particles cross the detection point, and decreasing again to a small concentration as the last particles cross the detection point. Consequently, the overall concentration profile will have the appearance of a Gaussian curve.
The approximately Gaussian concentration profile of a plug of material can be modeled using the Green's function solution to the one-dimensional diffusion equation. The Green's function solution is expressed as:
      G    ⁡          (              x        ,        t        ,        D            )        =            1                        4          ⁢          π          ⁢                                          ⁢          Kt                      ⁢          exp      (                        -                      x            2                                    4          ⁢          Kt                    )      where x is the distance from the centroid of the Gaussian curve, t is time from the insertion point to the detection point, and K is the Taylor-Aris dispersion coefficient. As the plug moves down the channel, the distance x is measured with respect to the centroid of the moving plug of material. In this model, at t=0 all of the material in the plug is at x=0. Consequently this model assumes that the plug is perfectly non-dispersed when it is introduced into the channel.
From the observed experimental Gaussian concentration profile of a plug of material dispersing in pressure-driven flow through a channel one can solve for the Taylor-Aris dispersion coefficient, K, and back out the molecular diffusivity, D, from the Taylor-Aris dispersion coefficient formula appropriate for the channel geometry. However, often the average velocity U used in calculating the Taylor-Aris dispersion coefficient must be determined based on estimated values of parameters such as viscosity and channel geometry. Further, using the Green's function without convolution requires one to assume that the plug is introduced to the channel in a perfectly non-dispersed plug.
Because the Taylor-Aris method requires making various assumptions in order to calculate the molecular diffusivity, the method is typically effective only for low velocity flow in channels with small radial dimensions. Nonetheless, the method is still often used today for measuring molecular diffusivities. Methods of calculating molecular diffusivity based on the measurement of a concentration profile at a single point in a channel have been previously described. One example is Michael S. Bello et al., “Use of Taylor-Aris Dispersion for Measurement of a Solute Diffusion Coefficient in Thin Capillaries,” Science, 266, 773–776 (1994). Measuring concentration only at a single point does not provide an accurate assessment of the change of a plug of material over the length of the channel. Also, these single point methods require extra steps to determine velocity.
Still others have calculated molecular diffusivities by measuring the concentration of a stream of solute in a microchannel. One example of such a method is discussed in Andrew E. Kamholz et al., “Optical Measurement of Transverse Molecular Diffusion in a Microchannel,” Biophys J, 80(4), 1967–1972 (April 2001). However, this requires a fairly large sample of the solute in order to create a constant stream and is not suitable when only small volumes of sample material are available. Further, the technique taught by Kamholz et al. requires complicated mathematical manipulation of data.
Thus, what is needed is a simple method for determining both the velocity and the molecular diffusivity simultaneously while using only a small volume of sample material.